Graphite and C60

As an illustration of results which can be derived using techniques described in the previous section, we present now some spectroscopic results on quasi two-dimensional graphite and the quasi zero-dimensional fullerene Ceo (for a recent review on fullerenes, see [9]). These results on graphite and C60 will help to understand the electron spectroscopy results on the quasi one-dimensional carbon nanotubes.

PES spectra of graphite [10] are shown in Fig. 2. For EB < 7.5 eV, the spectral weight is mainly caused by the density of occupied n-states. Here we see a linear increase of intensity with increasing binding energy followed by a maximum at EB ~ 3 eV in agreement with LDA band structure calculations [12]. The spectral weight at higher binding energies with a peak at 7.5 eV is mainly caused by the density of states of the occupied a-bands.

In Fig. 3 we show ARPES and ARIPES data of occupied and unoccupied bands of graphite, respectively, together with LDA band-structure calculations [1]. We focus our discussion primarily on the n-bands. A simple tight-binding calculation of a single graphene sheet yields a total width of the n-bands of 670 were 70 is the ppn hopping integral between two carbon sites.

Fig. 2. Photoemission spectra of graphite and annealed multiwall carbon nanotubes recorded with a photon energy of 40.8 eV. (from [10])

This calculation also yields a flat band region near the M-point in the Bril-louin zone, near EB = ±70 for both the occupied n and unoccupied n* states. In a real graphite crystal, there is a small splitting of the bands due to the weak van der Waals interaction between the graphene layers. The flat band region causes a maximum (van Hove singularity) in the density of states [12], which is also detected in the PES spectrum at EB & 3eV (see Fig. 2). The binding energy of this maximum, the energy separation of the two flat band regions E = 5 eV = 270 and the total width of the n bands W = 15 eV = 670 yields a hopping integral for the n electron between C atoms of 70 ~ 2.5eV. However, a somewhat larger value, 3.1eV, is needed to reproduce the fine details of the graphite band structure [13]. The gap between the a and the a* band is about 9eV and the total width of the a bands is of the order of 40 eV.

Figure 4 displays a PES spectrum of solid C60, which is quite different from that of graphite [14]. What is striking is the sharp and well separated features which correspond to the highly degenerate molecular levels of C60 [15]. Since the interaction between the C60 molecules is about 50 times smaller than the intra-ball hopping integral, the broadening of the molecular levels in the solid is rather small. This is in agreement with band structure calculations which yield a width of the molecular-level derived bands of about 0.5 eV [16]. A further broadening observed in the PES spectra is due to excitations of phonons and probably also due to a multiplet splitting. The features for Eb < 5 eV can be interpreted by molecular levels having predominantly n character. At higher binding energies, also a molecular orbitals contribute to the spectral weight. The comparison of the PES spectra of graphite and C60 clearly demonstrates the difference between a quasi two-dimensional and a quasi zero-dimensional system. In the former case, wide n-bands are observed, while in the latter, the density of states is dominated by only slightly broadened molecular levels.

Fig. 4. Left-hand panel: Photoelectron spectrum of solid C60. The excitation energy was hv = 21.2 eV. Right-hand panel: C 1s excitation spectrum of solid C60 measured with EELS (from [14])

Fig. 3. ARPES and IPES of graphite. The IPES data have been obtained from highly oriented pyrolytic graphite. Therefore, the rM and rK directions are indistinguishable. Solid line: band-structure calculations [11]. (from [1])

Fig. 4. Left-hand panel: Photoelectron spectrum of solid C60. The excitation energy was hv = 21.2 eV. Right-hand panel: C 1s excitation spectrum of solid C60 measured with EELS (from [14])

In Fig. 5 we show XES spectra of graphite and Ceo, which also should provide information on the occupied density of states [17]. The shape of the XES spectrum of graphite is rather similar to that of the PES spectrum shown in Fig. 4. For solid C60, more pronounced features appear near 280 and 282 eV

-Nanotubes

- Graphite

C«,

>s\ !

\\ ;

Energy [eV]

270 275

Energy [eV]

Fig. 5. XES of carbon nano-tubes (heavy line), graphite (light line) and solid Côo (dotted curve). (from [17])

due to the quasi molecular electronic structure of this material. This is also observed for the unoccupied states which are probed by transitions from the C 1s core level to the unoccupied states.

An EELS spectrum of graphite is presented in Fig. 6f. One maximum is observed at 285 eV, corresponding to transitions into unoccupied n* states [18]. The width of this resonance is considerably reduced compared to the width of the unoccupied n* band. This comes from an excitonic enhancement of the spectral weight at the bottom of the n* band [19]. Above 291 eV, core-level excitations to the unoccupied a* bands take place. In Ceo (Fig. 4, right-hand panel) [14], four features corresponding to unoccupied n* molecular levels are present below the a* onset at 291 eV.

Finally we come to valence band excitations recorded by EELS. In Fig. 7 (left figure) we show the loss function (3) of graphite together with the real and the imaginary part of the dielectric function, £1(w, q) and e2(w, q), derived from a Kramers-Kronig analysis [5]. The data were taken at a small wave vector parallel to the graphene sheets (small scattering angle), q = 0.1 A"1, which is much smaller than the Brillouin zone. In this case the data are comparable to optical data derived previously from reflectivity measurements [20]. In this geometry, the in-plane component of the graphite dielectric tensor is probed. Its imaginary part, e2, which is related to absorption, shows a Drude-like tail at low energy due to the small concentration of free carri-

290 295 Energy (eV)

Fig. 6. C 1s and K 2p excitation spectra of: (a) intercalated graphite KC8; intercalated SWNTs with C/K ratio of (b) 7± 1; (c) 16 ± 2, and (d) 34 ± 5; (e) pristine SWNTs, and (f) graphite. (from [18])

ers (electrons and holes). The tail is followed by an oscillator resonance at 4 eV. The latter corresponds predominantly to a n-n* transition between the flat band regions at the M-point [21]. The second peak in e2 near 12 eV is predominantly caused by a-a* transitions. The n-resonance at 4eV causes a zero-crossing of ei near 6eV where e2 is small and, therefore, the loss function (3) shows a maximum there, i.e. a plasmon is observed. Since this plasmon is related to a n-n* interband transition, we call this plasmon an interband plasmon or a n-plasmon. When turning to higher wave vectors, i.e., going from vertical transitions to non-vertical transitions, the energy of the transitions from n to n* states increases and therefore the n-plasmon shows dispersions to higher energies.

The second peak in the loss function of graphite at 27 eV is caused by the zero-crossing of e1 near 25 eV. This zero-crossing is related both to the number of valence electrons and to the energy of the n-n* and, predominantly, of the a-a* transitions. Since this plasmon involves all the valence electrons, it is called the n + a plasmon. Using a simple Lorentzian model for the dielectric function, e(w) = 1 +

Fig. 7. Electron energy loss function Im[—1/e], and the real (ei) and imaginary (£2) parts of the dielectric function of graphite and solid C60. (from [5] and [22])

Fig. 7. Electron energy loss function Im[—1/e], and the real (ei) and imaginary (£2) parts of the dielectric function of graphite and solid C60. (from [5] and [22])

yields for the plasmon energy E'p = y E0 + Ep where E0 = hu0 is the oscillator energy, 7 is a damping frequency, and Ep = h^ne2/me0 is the free-electron plasmon energy, with the valence electron density n and effective mass m. This relation clearly shows that the dispersion of the n + a plasmon as a function of the wave vector is not only determined by the free electron plasmon dispersion Ep (q) = Ep (0) + ah2q2/m with a = (3/5)EF/Ep (0), but also by the wave-vector dependence of the a and the n oscillators, i.e., mainly the a-a* transition.

For solid C60 the loss function (see Fig. 7, right-hand figure) [22] shows differences when compared to graphite. There is a gap of 1.8 eV followed by several n-n* transitions between the well-separated molecular levels having n-electron character. Following the same argumentation as for graphite, these n oscillators now cause several n plasmons in the energy range 1.8 to 6eV, although £\ does not vanish in that interval. The last peak at 5eV has the highest intensity because e2 is smaller there. In addition, there are several a-a* transitions which cause an almost zero-crossing of £\ near 22 eV, leading to a wide maximum in the loss function, i.e. the n + a-plasmon [23]. As mentioned before, the molecular orbital levels are only slightly broadened by the interaction between the Ceo molecules, which implies that the n-electrons are strongly localized on the molecules. Therefore, there is almost no wave-vector dependence of the energy of the n-n* transitions, and consequently no dispersion of the n-plasmons is detected [22].

3 Occupied States of Carbon Nanotubes

To illustrate the problems of the surface contamination encountered in PES spectroscopy of carbon nanotubes, we show in Fig. 8 wide-range X-ray induced PES [24] of a film of purified SWNTs. The dominant feature at 284.5 eV corresponds to the C 1s level. The binding energy is close to values observed in graphite and fullerenes. In addition, strong contaminations with O, N and Na are detected which show up by the O 1s, O:KLL (Auger peak), N 1s and Na 1s peaks, respectively. After annealing the sample in UHV at 1000°C, most of these contaminants have disappeared. However, small Ni and Co contamination from the catalyst remains, as indicated by the low intensity Ni 2p and Co 2p lines.

In Fig. 9 we show PES spectra of annealed SWNT films taken with the photon energies hv = 21.2 and 40.8 eV. For the higher energy photons, the PES spectrum is quite similar to the graphite spectrum shown in Fig. 2. A peak is observed at EB = 2.9 eV which can be assigned to a large density of states of the n states near EB = y0 (Fig. 11). A further peak is observed at Eb = 7.5 eV caused by a large density of a states. For hv = 21.2 eV, the C 2s states at lower EB are more pronounced.

T 0:KLL N1s ,

C1s

1200 1000 800 600 400 200 0 Binding energy (eV)

1200 1000 800 600 400 200 0 Binding energy (eV)

Fig. 8. X-ray induced photoelectron spectra of purified single-wall carbon nanotubes (a) without annealing and (b) after annealing at 1000° C (from [24])

Fig. 9. Photoelectron spectra of annealed SWNT using a photon energy of hv = 21.2 eV (•, shifted vertically for more clarity) and 40.8 eV (o). The inset gives results for low binding energies on an expanded scale (from [24])

Binding energy (eV)

Fig. 9. Photoelectron spectra of annealed SWNT using a photon energy of hv = 21.2 eV (•, shifted vertically for more clarity) and 40.8 eV (o). The inset gives results for low binding energies on an expanded scale (from [24])

Of particular interest is the question whether a Fermi edge can be observed from the metallic tubes. For one-dimensional systems, important deviations from Fermi liquid theory or even its breakdown is expected. Theoretical works predict a Luttinger liquid model in which the spectral weight close to the Fermi level should be suppressed [25]. Recent transport measurements indicate a Luttinger-liquid behavior in metallic SWNTs [26] and MWNTs as well [27]. The parameters derived from these measurements would lead to a considerable suppression of the spectral weight near EF. In the experiment, a clear Fermi edge is observed which, however, is probably due to small metallic catalyst particles (Ni and Co) that are also detected in the core level spectra. Further information on the occupied density of states of SWNTs is obtained from XES spectra, which are shown in Fig. 5, along with the spectra for C60 and micro-crystalline graphite for comparison [17]. The overall shape of the spectral profile is intermediate between graphite and C60, in particular close to 282 eV. A closer inspections, however, yields better agreement between graphite and SWNTs than between C60 and SWNTs. This observation indicates that the nanotubes are closer to graphite than to C60 fullerite.

Some information on the band dispersion in SWNTs can be obtained from RIXS [17]. The corresponding spectra are shown in Fig. 10. In the spectra excited at 285eV (bottom curve), the broad feature at 270eV emission energy, labeled B, is due to emission from the lowest two occupied bands at the K point, corresponding to states around 13 eV below EF in the band structure shown in Fig. 3. The reason for this assignment comes from the fact that the lowest excited states just above EF, which can be reached with photons of 285eV, are also located at the K point, and because the wave vector is conserved. Similarly, the peak labeled A can be assigned to occupied n-bands

265 270 275 280 285 290

Energy (eV)

Fig. 10. Resonant inelastic X-ray scattering spectra of single-wall carbon nanotubes (heavy line) and non-oriented graphite (light line). The photon energies (indicated by the arrows) used for the excitation are 285.0, 285.8, 286.2, 286.6 and 289.0 eV (from bottom to top). The raw data were corrected for the incoherent scattering and diffuse reflection (from [17])

265 270 275 280 285 290

Energy (eV)

Fig. 10. Resonant inelastic X-ray scattering spectra of single-wall carbon nanotubes (heavy line) and non-oriented graphite (light line). The photon energies (indicated by the arrows) used for the excitation are 285.0, 285.8, 286.2, 286.6 and 289.0 eV (from bottom to top). The raw data were corrected for the incoherent scattering and diffuse reflection (from [17])

close to the K point. With increasing excitation energy the feature A moves to lower energy both for graphite and SWNTs, which is expected for a dispersive ^-band (Fig. 3). Regarding the structure of SWNTs, it is not surprising that the occupied density of states and the data for the ^-band dispersion of SWNTs is very close to that of graphite. Each SWNT can be thought of as a single graphene layer that has been wrapped into a cylinder with a diameter of about 1.5 nm.

In Fig. 2 we show PES data for annealed MWNTs together with data for graphite. Again the difference between the two materials is rather small. Since the MWNTs consist of several graphene layers wrapped up into concentric layers, this close similarity of the electronic structure of the two systems is expected.

Fig. 11. (a) Calculated electronic density of states (states/eV/atom) of three different nanotubes derived from tight binding (70 = 2.75 eV): (12,8) (full curve), (10,10) (dashed curve) and (16,4) (dotted curve). (b) Arithmetic average of the densities of states of the 7 nanotubes with indices (10,10), (11,9), (12,8), (13,7), (14,6), (15,5), and (16,4). These calculations involve both n and a states [30]

Fig. 11. (a) Calculated electronic density of states (states/eV/atom) of three different nanotubes derived from tight binding (70 = 2.75 eV): (12,8) (full curve), (10,10) (dashed curve) and (16,4) (dotted curve). (b) Arithmetic average of the densities of states of the 7 nanotubes with indices (10,10), (11,9), (12,8), (13,7), (14,6), (15,5), and (16,4). These calculations involve both n and a states [30]

On the other hand, a closer look at the structure would indicate differences in the electronic structure between graphite and carbon nanotubes. Firstly, there is a quantum confinement effect perpendicular to the axis of the tube. The wave function of the rolled-up graphene sheet has to satisfy periodic boundary conditions around the circumference. Hence the component of the Bloch wave vector perpendicular to the axis, k±, can only assume discrete values. This means also that discrete energy values for the k± values are allowed [28]. Therefore, the electronic structure might be regarded as molecular-like in the circumferential direction. This molecular electronic structure for k perpendicular to the axis leads to van Hove singularities in the density of states [29], not encountered in that of graphite. The calculated density of states of the semiconducting (12,8) and the metallic (10,10) and (16,4) nanotubes are shown in Fig. 11. The above-mentioned van Hove singularities are clearly recognized.

Regarding the PES data shown in Fig. 9 that were recorded with an energy resolution of 0.1eV, one does not see the expected occupied van Hove singularities that should appear near 0.3, 0.6 and 1.2 eV for the semiconducting SWNTs, and at 0.9 eV for the metallic tubes. These singularities have clearly been observed on individual nanotubes by scanning tunneling spec-troscopy [31,32] and also through resonant Raman scattering [33,34]. The sample used in Fig. 9 comprised different kinds of SWNTs, including a distribution of diameters and chiralities. It is now well established from various diffraction experiments that a rope of SWNTs can mix tubes with different helicities [35], and the nanotube diameters may vary from one rope to the other within some limits [36]. Taking an average of the densities of states of different nanotubes brings them closer to graphite. This effect is clearly shown in Fig. 11b, which represents the arithmetic mean of the densities of states of 7 nanotubes with wrapping indices (10,10), (11,9) ... (15,5), (16,4) and diameters 1.4 ±0.03 nm. The averaging process and the small broadening of the densities of states used in the calculations (0.1eV) washed out most of the one-dimensional van Hove singularities of the individual nanotubes. In the experiment, the peaks are possibly broadened by phonon excitations, as is also observed in PES spectra of fullerenes. Defects and correlation effects may also broaden the spectral weights related to these singularities.

In MWNTs, which have diameters of the order of 10 nm, the van Hove singularities should appear at considerably lower energy [27]. In this case it is clear that they cannot be detected in the PES spectrum. In effect, Fig. 2 show experimental PES data of annealed MWNTs together with data of graphite. Again the difference between the two materials is rather small. Since the MWNTs consist of several graphene layers wrapped up into concentric layers, this close similarity between the electronic structures of the two systems is expected.

4 Unoccupied States of Carbon Nanotubes

There is only little information presently available from electron spectro-scopies on the unoccupied density of states of carbon nanotubes. Only core level C 1s EELS [18] and XAS [17] data on SWNTs have been reported. In Fig. 6e we show C 1s excitation spectra recorded by EELS in transmission. As in graphite (see Fig. 6f) the first peak at 285 eV corresponds to transitions into unoccupied n*-states while above 292 eV mainly unoccupied a*-states are detected. Since in graphite the maximum of the density of states 2.5 eV above EF is strongly shifted to lower energy due to the interaction with the core hole, the maximum appears at 1 eV above threshold. Probably the same happens in the SWNT spectra. The unoccupied van Hove singularities should appear at 0.3, 0.6 and 1.2 eV above threshold for the semiconductors, and at 0.9 eV above threshold for the metallic SWNTs. However, in this energy range the spectral weight is dominated by the peaked n* density of states, which is shifted to lower energies. Therefore, no details of the unoccupied density of states near the Fermi level could be detected in the EELS spectra, although the energy resolution in these experiments was 0.1 eV. The origin of the small peak at 287 eV which does not appear in all samples, is at present unclear.

5 Excited States on Carbon Nanotubes

In this section we review valence-band excitations recorded by momentum dependent EELS measurements on transmission [37]. In Fig. 12 we show the loss functions of purified SWNTs for various momentum transfers, q. The

Fig. 12. The magnitude of the loss function of purified singlewall carbon nanotubes for various momentum transfers q. The inset shows the loss function over a larger energy range for q = 0.15 A-1 (from [37])

Fig. 12. The magnitude of the loss function of purified singlewall carbon nanotubes for various momentum transfers q. The inset shows the loss function over a larger energy range for q = 0.15 A-1 (from [37])

wide energy-range loss function for q = 0.15 A-1 is shown in the insert. Similar to graphite, a n-plasmon is detected at 5.2 eV and a n+<r-plasmon appears at 21.5 eV. The same plasmon energies have been reported from EELS spectra recorded using a transmission electron microscope [38]. The reason for the lower energy of the n + a-plasmon compared to that of graphite partly comes from the fact that the carbon density, and therefore the electron density which determines the free electron plasmon energy, Ep, is smaller (see Sect. 2). But also, unlike the case of planar graphite, the wave-vector transfer q is tangential to some parts of the nanotube and normal to other parts, simply due to curvature. The anisotropy of the graphene layer is responsible for a down shift of the plasmon energy as compared to graphite. This interpretation is supported by theoretical calculations of EELS spectra based on atomic, discrete-dipole excitations in hyper-fullerenes [39]. The influence of the layer anisotropy on the plasmon excitation spectrum has clearly been demonstrated in the case of multi-shell systems [40,41].

Coming back to (1), the dielectric function of the nanotube sample is best defined for a crystalline arrangement of identical tubes, such as realized in a close-packed rope. A Lindhardt-like formula for e(u, q) can then be determined and the plasmons come out of the calculations as the zeros of the real part of it. Such calculations have been performed for the n electrons described by a tight-binding Hamiltonian, while ignoring the inter-tube coupling [42,43]. When the electric field is parallel to the nanotube axis, the real part of the long-wavelength dielectric function, e1(u), is shown to vanish for a special energy hu close to 270 ~ 6 eV [42]. The corresponding longitudinal mode is the n-plasmon. Its excitation energy is independent on the tube diameter and chirality [44].

The momentum-dependent measurements (Fig. 12) show a strong dispersion of the n-plasmon, similar to that of graphite. As discussed in Sect. 2, this plasmon is related to a n-n* interband transition near 5 eV, and therefore the dispersion indicates dispersive bands of delocalized n-electrons as in graphite. This dispersion can only occur along the SWNT axis and therefore the n-plasmon at 5.2 eV is related to a collective oscillation of n-electrons along the tube. On the other hand, the low-energy peaks at 0.85, 1.45, 2.0, and 2.55 eV show no dispersion as a function of the momentum transfer, similar to the n-plasmons in solid C60 (see the discussion in Sect. 2). This indicates that the low-energy maxima are related to excitations of localized electrons. It is tempting to attribute these excitations to collective excitations around the circumference of the tubes. The interband transitions which cause these peaks are then related to transitions between the van Hove singularities shown in Fig. 11a. In this view, the lowest two peaks appear as due to semiconducting tubes while that at 2.0 eV can be assigned to metallic tubes. This is also supported by regarding the data of e2 derived from Kramers-Kronig analysis (not shown). Peaks are detected at 0.65, 1.2 and 1.8 eV, which are close to the energy distance between van Hove singularities of semiconducting and metallic SWNTs in the tube diameter range relevant here. Measurements of the occupied and unoccupied density of states by scanning tunneling spec-troscopy are in agreement with these results [31,32]. From the intensity of the transitions, one can immediately conclude that about two third of the tubes are semiconducting while one third is metallic. Later on, transitions between the van Hove singularities have also been observed at 0.68, 1.2 and 1.8 eV with higher energy-resolution by optical absorption measurements [45,46], which agree very well with the positions of the e2(u) peaks.

The e2 data (not shown) or the optical absorption data are in a first approximation a measure of the joint density of states averaged over all existing SWNT structures existing in the sample. From band-structure calculations, it was predicted that the gap in the semiconducting samples or generally the energy distances between the van Hove singularities are inversely proportional to the diameter of the SWNTs [47,48]. This was confirmed by scanning tunneling microscopy and spectroscopy measurements [31,32,49]. Then the energies of the transitions yield information on the mean diameter of the tubes in the sample. The fact that well-defined peaks are observed in the loss function or in the optical absorption indicates a narrow diameter distribution, as otherwise the energetically different interband transitions would wash out the maxima like in Fig. 11b.

On the other hand, recent high-resolution optical measurements have detected a fine structure of the absorption lines [46]. This is illustrated in Fig. 13 where typical optical absorption data of the first three low-energy lines are shown as a function of the synthesis temperature. Peaks A and B correspond again to transitions between van Hove singularities in semiconducting tubes, while peak C corresponds to an excitation of metallic tubes. With increasing growth temperature, the mean energy of the three peaks are shifted to lower energies, i.e. the mean diameter of the tubes increases with increasing synthesis temperature. Moreover, the peaks clearly show a fine structure which is more pronounced in the high-energy peaks (C). The energy position of the vast majority of the sub peaks remained constant within the resolution

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